We consider the problems of estimation and optimization of utility-based shortfall risk (UBSR), which is a popular risk measure in finance. In the context of UBSR estimation, we derive a non-asymptotic bound on the mean-squared error of the classical sample average approximation (SAA) of UBSR. Next, in the context of UBSR optimization, we derive an expression for the UBSR gradient under a smooth parameterization. This expression is a ratio of expectations, both of which involve the UBSR. We use SAA for the numerator as well as denominator in the UBSR gradient expression to arrive at a biased gradient estimator. We derive non-asymptotic bounds on the estimation error, which show that our gradient estimator is asymptotically unbiased. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) algorithm for UBSR optimization. Finally, we derive non-asymptotic bounds that quantify the rate of convergence of our SG algorithm for UBSR optimization.

在金融领域中，我们考虑了效用损失风险（UBSR）的估计和优化问题，推导了经典样本平均逼近（SAA）的UBSR的均方误差的非渐近界限，以及在平滑参数化下UBSR梯度的表达式，将它用于UBSR优化的随机梯度算法中，推导了非渐近界限以衡量该算法的收敛速率。

基于效用的差额风险优化：一个非渐近视角